Nonlinear Control Design for a Multi-Terminal VSC-HVDC System

Yijing Chen1, Jing Dai1, Gilney Damm2, Francoise Lamnabhi-Lagarrigue1

Abstract— This paper presents a nonlinear control strategybased on dynamic feedback linearization control theory and abackstepping-like procedure for a multi-terminal voltage-sourceconverter based high-voltage direct current (multi-terminalVSC-HVDC) system. The controller is able to provide asymp-totic stability for the power transmission system with multipleterminals. In particular, it is shown that the zero dynamics(mostly representing the DC network) is exponentially stable.These results are obtained by a stability proof for the wholesystem under the proposed controller, and its performance isillustrated by computer simulations.

I. INTRODUCTION

With the development of wind, solar and other renewableenergy sources, there is an urgent need to integrate thesedecentralized and relatively small-scale power plants intothe grid in an economical and environmentally friendly way.Furthermore, the increase in electricity demand requires theexpansion of grid capacity. However, it is hard to upgrade thegrid with overhead AC lines, which occupy large transmis-sion line corridors. For both cases, Voltage-Source Converterbased High-Voltage Direct Current (VSC-HVDC) multipointnetworks could be a good solution.

At present, over 90 DC transmission projects exist inthe world, the vast majority for point-to-point two-terminalHVDC transmission systems [1] and only two for multi-terminal HVDC (MTDC) systems. The traditional two-terminal HVDC transmission system can only carry outpoint-to-point power transfer. As the economic developmentand the construction of the power grid require that the DCgrid can achieve power exchanges among multiple powersuppliers and multiple power consumers, MTDC systemsdraw more and more attention. As a DC transmissionnetwork connecting more than two converter stations, theMTDC transmission system offers a larger transmissioncapacity than the AC network and provides a more flexible,efficient transmission method. The main applications ofMTDC systems include power exchange among multi-points,connection between asynchronous networks, and integrationof scattered power plants like offshore renewable energysources such as wind farms and solar plants.

A large amount of research on two-terminal VSC-HVDCcontrol has been carried out [2], [3], [4], [5]. In [2], anequivalent continuous-time averaged state-space model is

This work is supported by WINPOWER project.1Y. Chen, J. Dai and F. Lamnabhi-Lagarrigue are with Laboratoire des

Signaux et Systemes (LSS), Supelec, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France (tel: +33 1 69 85 17 77, e-mail: yijing.chen@lss.supelec.fr,jing.dai@supelec.fr, francoise.lamnabhi-lagarrigue@lss.supelec.fr).

2G. Damm is with Laboratoire IBISC, Universite d’Evry-Val d’Essonne,40 rue du Pelvoux, 91020 Evry, France (e-mail: gilney.damm@ibisc.fr).

presented and a robust DC-bus voltage control scheme isproposed highlighting the existence of fast and slow dy-namics that can be associated to the inner current controlloop and outer DC-bus voltage control loop. Reference [4]proposes a control strategy under balanced and unbalancednetwork conditions, which contains two sets of controller:a main controller in the positive dq frame using decouplingcontrol, and an auxiliary controller using coupling control.However, the above mentioned controllers were designed fora standard two-terminal VSC-HVDC system, not for multi-terminal VSC-HVDC system. In [6], [7], control strategiesof multi-terminal VSC-HVDC systems were investigated.Reference [6] uses a droop control scheme to control theDC voltage. Reference [7] proposes a scheme for controllingand coordinating the VSC and sharing the power amongthe connected AC areas. However, the previously mentionedarticles came from the power systems community and, as aconsequence, they did not provide stability proofs.

In the present paper, a control strategy is formally de-signed with its mathematical stability analysis for a multi-terminal HVDC system. The controller is based on feedbacklinearization control theory (see [8], [9]) and it is developedby following a backstepping-like procedure (see [10], [11],[12], [13]). This controller assures asymptotic stability forthe power transmission system with multiple terminals. Ina second step, the behavior of the internal states of thesystem (known as the zero dynamics [14]), representing thetransmission network, is analyzed and the zero dynamics isshown to be exponentially stable. It can then be seen thatthe MTDC system is asymptotically stable.

This paper is organized into five sections. In Section II,a dynamic multi-terminal VSC HVDC model is given. InSection III, a feedback control law is developed. Simulationresults are presented in Section IV. Conclusions are drawnin Section V.

II. MODELING OF A MULTI-TERMINAL VSC-HVDCSYSTEM

This section introduces the state-space model of a multi-terminal VSC-HVDC system established in the synchronousdq frame, which allows for a decoupled control on the activeand the reactive power, with the high-frequency pulse widthmodulation (PWM) characteristics of the power electronicsneglected. Only the balanced condition is considered in thispaper, i.e. the three phases have identical parameters andtheir voltages and currents have the same amplitude whilephase-shifted 120 between themselves.

A converter of a multi-terminal VSC-HVDC system isshown in Fig. 1, where vli is the voltage of AC area i, Rli

2013 European Control Conference (ECC)July 17-19, 2013, Zürich, Switzerland.

978-3-952-41734-8/©2013 EUCA 3536

and Lli represent series connected phase reactors, ili is theAC current through the phase reactors, vi is the voltage onthe AC side of the converter, Ci is the DC capacitor, ici anduci are the DC bus current and the DC voltage, Rci and Lci

are the transmission cable resistance and inductance, and iiis the current on the DC side of the converter.

Fig. 1. One terminal in a multi-terminal VSC transmission system.

A. AC network

On the AC side of the converter station, the Kirchhoffvoltage law leads to the system expressed in dq synchronousreference frame rotating at the pulsation ωi:

vlid −Rliilid − Llidiliddt

+ ωiLliiliq − vid = 0

vliq −Rliiliq − Llidiliqdt− ωiLliilid − viq = 0

By using PWM technology, the amplitude of the converteroutput voltage vid and viq are controlled by the modulationindex Mdi and Mqi as:

vid =Mdi

2uci

viq =Mqi

2uci

By neglecting the resistance of the converter reactor andswitching losses, the instantaneous active power and thereactive power on the AC side of the converters can beexpressed as:

Pli =3

2(vlidilid + vliqiliq) (1)

Qli =3

2(vliqilid − vlidiliq) (2)

B. DC line

By applying the Kirchhoff voltage and current laws to theDC circuit, the DC side of the converter is modeled by:

ducidt

= − 1

Ciici +

1

Ciii

dicidt

=1

Lciuci −

Rci

Lciici −

1

Lciucc

C. AC-DC power coupling

Because of the active power balance on both sides of theconverter, we have uciii = viAiliA+viBiliB+viCiliC . Thus,ii can be expressed as:

ii =3

4(Mdiilid +Mqiiliq)

A radial structure is chosen for the interconnection of theN terminals, as shown in Fig. 2, which is represented as:

duccdt

=1

Cc

N∑k=1

ick

……

VSC

1 AC

1

VSC

N AC

N

Fig. 2. The interconnection between N terminals.

D. Global model

The full state-space model of the N -terminal VSC-HVDCsystem is written as:

diliddt

= −Rli

Lliilid + ωiiliq −

1

LliMdi

uci2

+1

Llivlid

diliqdt

= −Rli

Lliiliq − ωiilid −

1

LliMqi

uci2

+1

Llivliq

ducidt

= − 1

Ciici +

1

Ci

3

4(Mdiilid +Mqiiliq)

dicidt

=1

Lciuci −

Rci

Lciici −

1

Lciucc

duccdt

=1

Cc(∑N

k=1 ick)

(3)where i = 1, · · · , N .

The global state-space model is summarized as:

• State variables: ilid, iliq , uci, ici, ucc.• Control variables: Mdi, Mqi.• External signals: vlid, vliq .• The dimension of the system (3) is 4N + 1.

III. CONTROL SCHEME

In this section, we present the detailed synthesis of thecontroller for one converter. The control objective is tomake the converter’s DC voltage uci and the reactive powerQli track their reference values u∗ci and Q∗

li. We use abackstepping-like procedure to design such a controller.

A. Controller synthesis

1) First step: The first step of the backstepping-likeprocedure consists in making the AC currents ilid and iliqfollow their reference trajectories i∗lid and i∗liq (yet to bedesigned in a second step), i.e. to eliminate the dq currenterrors ilid and iliq where ilid = ilid−i∗lid and iliq = iliq−i∗liq.

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Following a feedback linearization procedure, we designthe control laws as

uid =diliddt

=diliddt

+di∗liddt

uiq =diliqdt

=diliqdt

+di∗liqdt

(4)

where uid and uiq are auxiliary inputs, which control inde-pendently ilid and iliq. By substituting (4) into the dq currentequations of the system (3), the control variables Mdi andMqi are expressed as:

Mdi =2Lli

uci(−Rli

Lliilid + ωiiliq +

1

Llivlid − uid)

Mqi =2Lli

uci(−Rli

Lliiliq − ωiilid +

1

Llivliq − uiq)

(5)

To assure that the errors ilid and iliq will converge to zero,the following augmented states are proposed:

ϕid = ilid˙ilid = −kidilid − λidϕid

(6)

ϕiq = iliq˙iliq = −kiq iliq − λiqϕiq

(7)

where kid, kiq , λid and λiq are positive constants. Bycombining (4), (6) and (7), we have:

uid = −kidilid − λidϕid +di∗liddt

uiq = −kiq iliq − λiqϕiq +di∗liqdt

(8)

2) Second step: The second step of the backstepping-likeprocedure determines the dq current reference values so thatthe converter tracks the reference values for the DC voltageand the reactive power.

By assuming a dq frame orientation such that vliq = 0, i∗liqcan be obtained directly from the reactive power’s reference:

i∗liq = −2

3

Q∗li

vlid(9)

As to i∗lid, it is used to keep the DC voltage constant atits reference point u∗ci. i

∗lid is calculated by the DC voltage

controller as follows. By substituting (5) into the thirdequation of (3), we have:

uci =−1

Ciici +

3

2

1

Ciuci[ilid(−Rliilid + vlid − Lliuid)

+ iliq(−Rliiliq + vliq − Lliuiq)] (10)

Then, by substituting (8) into (10), the DC voltage dynamicsis given by:

uci =−1

Ciici +

3

2

1

Ciuci[ilid(−Rliilid + vlid + Llikidilid

+ Lliλidϕid − Llidi∗liddt

) + iliq(−Rliiliq + vliq

+ Llikiq iliq + Lliλiqϕiq − Lli

di∗liqdt

] (11)

To maintain the DC voltage uci at its set value, the desireddynamics of voltage error uci is expressed as:

ϕci = uci˙uci = −kciuci − λciϕci (12)

where uci = uci − u∗ci. The above equation can also bewritten as:

uci = −kciuci − λciϕci + u∗ci (13)

Since the desired values for u∗ci, Q∗li are constant,

du∗cidt

anddi∗liqdt

are zero.

By combining (11) and (13),di∗liddt

is deduced as:

di∗liddt

=− 2

3

uciilid

Ci

Lli(−kciuci − λciϕci +

iciCi

) +uci2Lli

iliqilid

Mqi

+ (−Rli

Lliilid + ωiiliq +

vlidLli

+ kidilid + λidϕid)

(14)

B. Stability studyTheorem 1: Under the control laws (5), (6), (7), (9), (12)

and (14), the multi-terminal VSC-HVDC system describedby (3) is asymptotically stabilized to their reference valuesu∗ci and Q∗

li. Furthermore, this result is independent of thenetwork parameters Lci, Rci, Cci.

Proof: In the considered case, all the N converterscontrol their DC voltages. Since it is desired to keep uci,ilid and iliq at their reference values u∗ci, i

∗lid and i∗liq, we

define the outputs of the system as:

y = [uci ilid iliq]T

To simplify our analysis, we first shift the referencevalues of the whole system to the origins by introducingthe following state variables:

x = [ilid iliq uci ici ucc]T

where ici = ici − i∗ci, ucc = ucc − u∗cc, and i∗ci and u∗cc arethe equilibrium values of ici and ucc. The new output errorvariables are defined as:

y = [uci ilid iliq]T

The system (3) can be expressed in terms of the new variablesas:

diliddt

= −Rli

Lliilid + ωiiliq −

1

2Lli(Mdiuci −M∗

diu∗ci)

diliqdt

= −Rli

Lliiliq − ωiilid −

1

2Lli(Mqiuci −M∗

qiu∗ci)

ducidt

= − 1

Ciici

+1

Ci

3

4(Mdiilid −M∗

dii∗lid +Mqiiliq −M∗

qii∗liq)

dicidt

=1

Lciuci −

Rci

Lciici −

1

Lciucc

duccdt

=1

Cc(∑N

k=1 ick)

(15)

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where i = 1, · · · , N and M∗di and M∗

qi are the equilibriumvalues of Mdi and Mqi.

In order to analyze the stability of the new system (15),we divide the state variables x into two parts:

η = [ici ucc]T

ξ = [ilid iliq uci]T

Then, system (15) can be considered as in the normal form:η = f1(η, ξ)

ξ = f2(η, ξ, u)(16)

withu = f3(η, ξ) (17)

where u = [Mdi Mqi].We see that if the output is identically zero, i.e. y ≡ 0,

then ξ ≡ 0, and the behavior of the system (16) is governedby the differential equation:

η = f1(η, 0) (18)

which is called the zero dynamics of the system.We now study the behavior of η and ξ. By combining (5)

(6) (7) and (12), the closed-loop error system can be writtenas:

ζ = Aζ

where ζ = [ϕid ilid ϕiq iliq ϕci uci]T and A =

diag(Aid, Aiq, Aci) with

Aid =

(0 1−λid −kid

)Aiq =

(0 1−λiq −kiq

)Aci =

(0 1−λci −kci

)It is easy to verify that matrix A is Hurwitz. Thus, ζ (henceξ) is exponentially stable under the proposed control law. Itremains now to study the behavior of the state variables ηas ξ converges to zero. In fact, when ξ = 0, η is governedby the following differential equation:[

˙ic1

˙ic2 · · · ˙icN ˙ucc

]T= B

[ic1 ic2 · · · icN ucc

]T

where B =

−Rc1

Lc10 . . . 0 − 1

Lc1

0 −Rc2

Lc2. . . 0 − 1

Lc2

......

......

...1Cc

1Cc

. . . 1Cc

0

.

Thus, the zero dynamics of the system (15) is:

f1(η, 0) = Bη (19)

To investigate the stability of (19), a Lyapunov functionV is chosen as:

V =

N∑k=1

Lck

Cci2ck + u2cc (20)

The derivative of V along the trajectories of (19) is givenby:

V =

N∑k=1

Lck

Ccick

˙ick + ucc ˙ucc

=

N∑k=1

Lck

Ccick(−

Rck

Lckick −

1

Lckucc) + ucc

1

Cc(

N∑k=1

ick)

= −N∑

k=1

Rck

Cci2ck ≤ 0 (21)

V is negative semidefinite. To find S = [η ∈ RN+1|V (η) =0], note that

V (η) = 0⇒ ick = 0, k = 1, · · · , N (22)

Hence, S = [η ∈ RN+1 |ick = 0, k = 1, · · · , N ]. Let η be asolution that belongs identically to S:

ick ≡ 0⇒ ˙ick ≡ 0⇒ ucc ≡ 0 (23)

Therefore, the only solution that can stay identically in Sis the trivial solution η ≡ 0. Thus, according to LaSall’stheorem and its corollary, the zero dynamics of the system(15) is asymptotically stable.

Therefore, the whole system (15) is asymptotically stabi-lized at (η, ξ) = (0, 0) under the proposed controller.

IV. SIMULATION RESULTS

The proposed controller is tested by computer simulationson a three-terminal VSC-HVDC system shown in Fig. 3. Allthree VSC converters operate in DC voltage control mode.The terminal parameter values are given in Table I. Wechoose ωi = 314, and vli = 230 kV. The feedback controlgains are chosen as kid = 100, λid = 100, kiq = 100,λiq = 100, kci = 25, λci = 5. The sequence of eventslisted in Table II is applied to the system. In addition, Ql1,Ql2 and Ql3 are set to zero to have a unitary power factor,which means that i∗l2q , i∗l2q and i∗l3q are zero.

AC

1

V

S

C

1

V

S

C

2

AC

2

VSC

AC 3

Fig. 3. A three-terminal VSC-HVDC transmission system.

Simulation results are shown in Figs. 4-9. The regulationresponse of each converter’s DC voltage is illustrated in Figs.4, 6, and 8 where the black curve represents the DC voltage’sreference value and the red one is the response. At the start

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Terminal Rli Lli Rci Lci Ci

1 13.79 Ω 31.02 mH 0.2085 Ω 2.4 mH 12 µF2 12.79 Ω 33.02 mH 0.2 Ω 1 mH 12 µF3 13.57 Ω 40.02 mH 0.235 Ω 3.5 mH 12 µF

TABLE IPARAMETER VALUES OF THE TERMINALS.

Time (s) Event0 u∗c1 = 101 kV, u∗c2 = 100 kV, u∗c3 = 99.8 kV1 u∗c1 = 101.2 kV, u∗c2 = 101 kV, u∗c3 = 99.9 kV4 u∗c1 = 102.2 kV5 u∗c2 = 102.0 kV6 u∗c3 = 100.9 kV

TABLE IISEQUENCE OF EVENTS APPLIED TO THE SYSTEM.

of the simulation, the converter work at their initial referenceDC voltage. After a step change in each DC voltage referencevalue at t = 1 s, the actual uci reaches the new u∗ci beforet = 2 s, as shown in Figs. 4, 6, and 8. At t = 4 s, only u∗c1has a step change, which uc1 follows before t = 5 s, as canbe seen in Fig. 4. From t = 4 s to t = 5 s, uc2 and uc3 keepunchanged and remain at their reference values, as shown inFigs. 6 and 8. After uc2 and uc3 have their reference valuesreset respectively at t = 5 s and t = 6 s, they attain theirnew reference values and have no effect on uc1.

0 1 2 3 4 5 6 7 818008

1801

18012

18014

18016

18018

1802

18022

180240010

5

Fig. 4. uc1 response.

Figs. 5, 7 and 9 illustrate each converter’s DC currentici. We see that, once the DC voltage reference value ischanged, the DC current is also changed and reaches thenew reference point. This shows the effectiveness of theDC voltage controller. A negative ici means that AC areai absorbs active power from the DC grid, while a positiveici means that AC area i injects active power into DC grid.

Simulation results (figures not shown here) show that theconverter quadrature current iliq is always very close to zerono matter how we change the DC voltage reference value.The reason is that the quadrature current is controlled by thereactive controller, which keeps iliq close to zero in order tohave a zero Qli.

0 1 2 3 4 5 6 7 82000

2500

3000

3500

4000

4500

5000

5500

6000

Fig. 5. ic1 response.

0 1 2 3 4 5 6 7 805555

1

15005

1501

15015

1502

150250010

5

Fig. 6. uc2 response.

0 1 2 3 4 5 6 7 801500

01000

0500

0

500

1000

1500

2000

2500

3000

3500

Fig. 7. ic2 response.

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0 1 2 3 4 5 6 7 806666

06668

1

16002

16004

16006

16008

16010010

5

Fig. 8. uc3 response.

0 1 2 3 4 5 6 7 807000

06000

05000

04000

03000

02000

01000

0

Fig. 9. ic3 response.

V. CONCLUSIONS

In this paper, a nonlinear controller is designed for amulti-terminal VSC-HVDC system. The proposed controllaw is based on dynamic feedback linearization strategy anda backstepping-like procedure. A detailed stability analysisby means of the zero dynamics approach for the nonlinearsystem shows that the MTDC system is asymptotically stableindependently of network parameters. Simulation resultsshow that the proposed control strategy is able to regulatethe DC-bus voltage with good dynamic performance.

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